Arithmetic

              Counting          

Counting  by .. 10's, 2's, 2 1/2's, ..  count in binary, and base 3 ; the Guess My Number game..  and Kaitlin's 6 "Magic number cards";  Jerry counts in many bases

Counting how many pieces of a size make the whole cake, to name a fraction of the cake in chapter 1 (infinite series), or cookie-sharing in chapter 2. This is a key idea which many students are not aware of and causes difficulty in all their math courses!

Counting the number of rows of hexagonal cells on a pineapple or the number of rows of seeds on a sunflower head to get Fibonacci numbers - an infinite sequence (see Map) and Chapter 7. 

Counting squares on a geoboard to find the area within a shape, and leads to a 3D graph and function.

Counting squares under a curve which leads to the integral. See chapter 13 and Geoff's and Grace's work.

Counting squares on graph paper to find patterns in the square numbers (see Tara's work).

Counting up, in looking at the differences in the output of a function (guess my rule- chapter 6). 

Counting small cubes that make cubes and pyramids, and other shapes (Genny doubles the size of a dog), in chapter 13 and see Sheri's work.

Counting the moves to interchange the pegs in the Shuttle Puzzle (or Peg Game) and get a function, in chapter 6.

Counting the minimum # of moves to move the discs in the Tower Puzzle in chapter 6. See also Sheri's work (see Map-functions, non-linear).

Counting the number of triangles and the number of edges in the Snowflake curve to obtain infinite series to find its area and perimeter. See chapter 4 and Emily's work.

Counting the number of images in the hinged mirrors to obtain a function (chapter 6 and Map).

Counting squares and cubes to find the infinite sequence of Surface area/Volume ratios of rods and why rodents are nocturnal animals (also see Map).

            Place-value ( 100's 10's 1's paper, Dienes blocks

            Patterns in addition- large numbers                

           Patterns in subtraction- large numbers,

                4-3..  3-4..     

                integers (by postperson stories-RBDavis; by patterns)           

            Patterns in multiplication- large numbers

                factoring

                prime numbers (Ramanujan, sieve of Aristosthanese (?sp))

                patterns on a 12-dot circle

            Patterns in division- large numbers

                ways to write division (see video)

                Cookie-sharing-> fractions

                remainders

                clock arithmetic

                12x13 in your head- (see The Math Program-mathletter #2 )

                continued fractions (Ramanujan)

                infinite continued fractions  (Ramanujan) 

                 
             Guess my number- leads to binary arithmetic

            Decimals-> infinite repeating decimals

            Ratio of perimeter of circle to its diameter (Kohler on map)

            Game of clues RBD- logic

Algebra

           
           identities for algebra

            Exponents

                identities for exponents

                identities for radicals

                fractional exponents

            Solving equations

                solving linear equations- by guessing, by balance-pictures (see Robert B. Davis, by transformations

                ways to solve quadratic equations-(see video Jenny & Don)

            Graphing functions, inverse functions (Descartes), slope and intercept

            Guess My Rule (guessing functions, W.W. Sawyer and function machines)

                Peg game ->quadratic function & inverse (start in 1st grade, just play the game, then to finding quadratic function)

               Tower puzzle -> exponential function & inverse (start in 1st grade, just play the game, then to finding exponential function)

            Word Problems (Polya and Descartes)

             Logarithms (Napier)

                identities for logs (Maya's work)

             Infinite geometric series (see video of Kirsten & Don, Alex work, Marina's -changing an infinite repeating decimal to a fraction using infinite series, video Marie&Don 8/3, Perimeter and area of                 snowflake curve by Emily, showing 3 cycles of an infinite repeating decimal by needlepoint)

            Infinite sequences-Marina's work; perimeter and area of  the snowflake curve by Emily,; Sheri finding the squareroot of 2; Jamie finding the sequence leading to the divine proportion

                notes on a piano (geometric mean)

                Fibonacci numbers and their ratios -> golden mean and spirals, video math in Nature

                 Sunflower head -> leads to Fibonacci numbers

                Sheri finding the ratio of the Volume of a pyramid to the volume of a cube (same base and height) - 4 ways

            Magic squares (see Ramanujan, the 15 puzzle)

            Group theory (see W.W. Sawyer)

Geometry

           
            geoboard (square- areas of shapes-Pick's Theorem)

            geoboard (circular- angles, inscribed, in a )

            Shapes- sides, angles -constructions with compass & straightedge

            rotogram (from Scotland)- sum of interior and exterior angles of a polygon

            Pantograph- making shapes similar- smaller and larger

            Group theory (see W.W. Sawyer)

            See W.W. Sawyer's 4 sheets

            Pythagoras (see ies Japan)

            Patterns in shapes

            Soma cube- 7 pieces (see Erin's 16 ways to make the cube)

            Problems from AHSME, Math Olympiads...

    Trigonometry